Theorem brcogw 4773

Description: Ordered pair membership in a composition. (Contributed by Thierry Arnoux, 14-Jan-2018.)

Assertion

Ref	Expression
brcogw	|- ( ( ( A e. V /\ B e. W /\ X e. Z ) /\ ( A D X /\ X C B ) ) -> A ( C o. D ) B )

Proof of Theorem brcogw

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpl1 990	. 2 |- ( ( ( A e. V /\ B e. W /\ X e. Z ) /\ ( A D X /\ X C B ) ) -> A e. V )
2		simpl2 991	. 2 |- ( ( ( A e. V /\ B e. W /\ X e. Z ) /\ ( A D X /\ X C B ) ) -> B e. W )
3		breq2 3986	. . . . . 6 |- ( x = X -> ( A D x <-> A D X ) )
4		breq1 3985	. . . . . 6 |- ( x = X -> ( x C B <-> X C B ) )
5	3, 4	anbi12d 465	. . . . 5 |- ( x = X -> ( ( A D x /\ x C B ) <-> ( A D X /\ X C B ) ) )
6	5	spcegv 2814	. . . 4 |- ( X e. Z -> ( ( A D X /\ X C B ) -> E. x ( A D x /\ x C B ) ) )
7	6	imp 123	. . 3 |- ( ( X e. Z /\ ( A D X /\ X C B ) ) -> E. x ( A D x /\ x C B ) )
8	7	3ad2antl3 1151	. 2 |- ( ( ( A e. V /\ B e. W /\ X e. Z ) /\ ( A D X /\ X C B ) ) -> E. x ( A D x /\ x C B ) )
9		brcog 4771	. . 3 |- ( ( A e. V /\ B e. W ) -> ( A ( C o. D ) B <-> E. x ( A D x /\ x C B ) ) )
10	9	biimpar 295	. 2 |- ( ( ( A e. V /\ B e. W ) /\ E. x ( A D x /\ x C B ) ) -> A ( C o. D ) B )
11	1, 2, 8, 10	syl21anc 1227	1 |- ( ( ( A e. V /\ B e. W /\ X e. Z ) /\ ( A D X /\ X C B ) ) -> A ( C o. D ) B )

Syntax hints:   -> wi 4   /\ wa 103   /\ w3a 968   = wceq 1343   E.wex 1480   e. wcel 2136   class class class wbr 3982   o. ccom 4608

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-co 4613

This theorem is referenced by: (None)